∫ 1 1 − x 2 arcsin x {\displaystyle \int {\frac {1}{{\sqrt {1-x^{2}}}\arcsin {x}}}}
u = arcsin x d u = 1 1 − x 2 d x {\displaystyle {\begin{aligned}u&=\arcsin {x}\\[2ex]du&={\frac {1}{\sqrt {1-x^{2}}}}\;dx\\[2ex]\end{aligned}}}
∫ ( 1 1 − x 2 1 arcsin x ) d x = ∫ 1 u d u = ln | u | + c = ln | arcsin x | + c {\displaystyle {\begin{aligned}\int ({\frac {1}{\sqrt {1-x^{2}}}}{\frac {1}{\arcsin {x}}})\;dx&=\int {\frac {1}{u}}\;du\\[2ex]&=\ln |u|+c\\[2ex]&=\ln |\arcsin {x}|+c\\[2ex]\end{aligned}}}
![{\displaystyle {\begin{aligned}u&=\arcsin {x}\\[2ex]du&={\frac {1}{\sqrt {1-x^{2}}}}\;dx\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/b4508e836235d223e2f8c84c555442628b62e838)
![{\displaystyle {\begin{aligned}\int ({\frac {1}{\sqrt {1-x^{2}}}}{\frac {1}{\arcsin {x}}})\;dx&=\int {\frac {1}{u}}\;du\\[2ex]&=\ln |u|+c\\[2ex]&=\ln |\arcsin {x}|+c\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/dc486b5a2bab32ba3d08998cd31ea83afe03b4c6)